Triangular prism

In geometry, a triangular prism is a prism with three side faces. This polyhedron has a triangular base as its faces, its copy obtained as a result of parallel transfer and 3 faces connecting the . A straight triangular prism has rectangular sides, otherwise the prism is called oblique .

A homogeneous triangular prism is a straight triangular prism with an equilateral base and square sides.

A prism is a pentahedron , in which two faces are parallel, while the normals of the other three lie in the same plane (which is not necessarily parallel to the bases). These three faces are parallelograms . All sections parallel to the bases are the same triangles.

Semi-Regular (Homogeneous) Polyhedron

A straight triangular prism is a semi - regular polyhedron or, more generally, a polyhedron, if the base is a regular triangle and the sides are squares .

This polyhedron can be considered as a truncated triangular osohedron represented by the Schleafli symbol t {2,3}. It can also be considered as a direct product of a triangle by a segment , which is represented as {3} x {}. The dual polyhedron of a triangular prism is a triangular bipyramid .

The symmetry group of a direct prism with a triangular base is D _{3h of} order 12. The rotation group is D ₃ with order 6. The symmetry group does not contain central symmetry .

Volume

The volume of any prism is equal to the product of the base area by the distance between the bases. In our case, when the base is triangular, you just need to calculate the area of the triangle and multiply by the length of the prism:

$V={\frac {1}{2}}bhl$ ${\ displaystyle V = {\ frac {1} {2}} bhl}$ $V = \frac{1}{2} bhl$ where b is the length of the side of the base, h is equal to the height of the triangle, and l is equal to the distance between the triangles.

Truncated Triangular Prism

A truncated straight triangular prism has one truncated triangular face ^[1] .

Faceting

There is a complete D _2h symmetry of the (removing a part of a polyhedron without creating new vertices, crossing the edges with a new vertex is not considered a triangular prism) . The resulting polyhedra have polyhedra with 6 faces in the form of an isosceles triangle , one polyhedron preserves the original upper and lower triangles, and one preserves the original squares. Two C _3v faceting symmetries have one base triangle, 3 faces in the form of lateral self-intersecting squares, and 3 faces in the form of isosceles triangles.

Convex	Cutting
Symmetry D _3h			Symmetry C _3v

2 {3} 3 {4}	3 {4} 6 () v {}	2 {3} 6 () v {}	1 {3} 3 6 () v {}	1 {3} 3 3 () v {}

Related polyhedrons and mosaics

Prism Family
Configuration	3.4.4	4.4.4	5.4.4	6.4.4	7.4.4	8.4.4	9.4.4	10.4.4	11.4.4	12.4.4	17.4.4	∞.4.4
Polygon
Mosaic

Convex Dome Family
n	2	3	four	five	6
Title	{2} \|\| t {2}	{3} \|\| t {3}	{4} \|\| t {4}	{5} \|\| t {5}	{6} \|\| t {6}
Dome	Diagonal dome	Three-Slope Dome	Four-Slope Dome		Six-domed (flat)
Related homogeneous polyhedrons	Triangular prism	Cuboctahedron	Rhombocubus octahedron	Rhomboicos dodecahedron

Symmetry Options

This polyhedron is topologically part of a sequence of homogeneous truncated polyhedra with vertex configurations (3.2n.2n) and having the symmetry [n, 3] of the Coxeter group .

Symmetry options * n 32 truncated mosaics: 3.2 n .2 n
Symmetry * n 32 [n, 3]	Spherical					Compact hyperbolic.		Paracom pact	Incompact hyperbolic.
Symmetry * n 32 [n, 3]	* 232 [2,3]	* 332 [3.3]	* 432 [4.3]	* 532 [5.3]	* 632 [6.3]	* 732 [7.3]	* 832 [8.3] ...	* ∞32 [∞, 3]	[12i, 3]	[9i, 3]	[6i, 3]
Truncated figures
	3.4.4	3.6.6	3.8.8	3/10/10					3.24i.24i	3.18i.18i	3.12i.12i
Divided figures
	V3.4.4	V3.6.6	V3.8.8	V3.10.10			V3.16.16	V3.∞.∞

This polyhedron is topologically part of a sequence of polyhedra with a vertex figure (3.4.n.4), which continues as refinements of the hyperbolic plane . These figures have mirror (* n32).

Symmetry options * n 42 extended mosaics: 3.4. n .4
Symmetry * n 32 [n, 3]	Spherical				Euclidean	Compact hyperbolic		Paracompact
Symmetry * n 32 [n, 3]	* 232 [2,3]	* 332 [3.3]	* 432 [4.3]	* 532 [5.3]	* 632 [6.3]	* 732 [7.3]	* 832 [8.3] ...	* ∞32 [∞, 3]
Figure
Configuration	3.4.2.4	3.4.3.4	3.4.4.4	3.4.5.4

Compound bodies

There are 4 homogeneous composite bodies of triangular prisms:

;
;
;
.

Honeycomb

There are 9 homogeneous honeycombs that include triangular prisms:

triangular prismatic honeycombs
trihexagonal prismatic honeycombs
truncated hexagonal prismatic cells
rhombo triangular prismatic honeycombs
hexagonal prismatic honeycombs
elongated triangular prismatic cells

Related Polyhedrons

The triangular prism is the first in the spatial series of . Each subsequent has a previous polyhedron as a vertex figure . discovered this series in 1900 as containing all kinds of faces of regular multidimensional polyhedra , containing all simplexes and orthoplexes ( regular triangles and squares in the case of a triangular prism). In the symbol −1 ₂₁ corresponds to a triangular prism.

in a space of dimension n
Space	Final						Euclidean	Hyperbolic
	3	four	five	6	7	eight	9	ten
Group Coxeter	E₃ = A₂A₁	E₄ = A₄	E₅ = D₅	E₆		E₈	E₉ = Ẽ₈ = E₈ ⁺	E₁₀ = T ₈ = E₈ ⁺⁺
Diagram Coxeter
	[3 ^−1,2,1 ]	[ ^3,2,2,1 ]	[3 ^1,2,1 ]	[3 ^2,2,1 ]	[3 ^3,2,1 ]	[ ^3,2,2,1 ]	[3 ^5,2,1 ]	[ ^3,2,2,1 ]
Order	12	120	192	51 840	2 903 040	696,729,600	∞
Graph							-	-
Designation	−1 ₂₁	0 ₂₁	1 ₂₁

Four-dimensional space

A triangular prism exists as a cell in a large number of four-dimensional , including:





				Cantellated tesseract

Notes

↑ William F. Kern, James R Bland, Solid Mensuration with proofs , 1938, p.81

Links

Weisstein, Eric W. Triangular prism on the Wolfram MathWorld website.
Interactive Polyhedron: Triangular Prism
surface area and volume of a triangular prism

[1] William F. Kern, James R Bland, Solid Mensuration with proofs , 1938, p.81